To be clear, I was not talking about the email address you use for your Avast account.
I was talking about the Google account on your phone, which you need in order for Google services (gmail/playstore/google+ etc.) to work. Usually you can not log out, but you can revoke access in 2-step verification, like I said.
So, I noticed that Avast web commands are not pushed to my phone after I revoke my phone’s access to Google services.